3) Consider the Euler angles involved in a 321 rotation from F_(W) to F_(0) (a) For the initial (F_(w)) and each intermediate coordinate system involved in this transformation determine which if any of the axes He in one of the three planes formed by pairs of axes of F_(B) Describe and sketch each case identified and explain why it is so (b) What is the value of thin the transformation?Explain. (c) Compare these rotations with the definitions of the aerodynamic angles atand B. and verify that the names of the Euler angles given for T_(BW) are comed. (d) Verify that the transformation matrix from wind to body axes is

(a) In a 321 rotation from to , the Euler angles involved are 321 degrees. This means that the rotation involves a 3-degree roll, a 2-degree pitch, and a 1-degree yaw. For the initial and each intermediate coordinate system involved in this transformation, we can determine which, if any, of the axes lie in one of the three planes formed by pairs of axes of . - For the initial coordinate system, none of the axes lie in one of the three planes formed by pairs of axes of .- For the first intermediate coordinate system (after the 3-degree roll), the x-axis lies in the xz-plane formed by the x and z axes of .- For the second intermediate coordinate system (after the 2-degree pitch), the y-axis lies in the yz-plane formed by the y and z axes of .- For the final coordinate system (after the 1-degree yaw), none of the axes lie in one of the three planes formed by pairs of axes of .(b) The value of the transformation is given by the rotation matrix that describes the 321 rotation from to . This matrix can be calculated using the Euler angles and the rotation matrices for each individual rotation (roll, pitch, and yaw). The resulting matrix will describe the transformation from the wind axes to the body axes.(c) The aerodynamic angles at A and B are defined as follows:- Angle A: The angle between the relative wind and the longitudinal axis of the aircraft.- Angle B: The angle between the relative wind and the lateral axis of the aircraft.By comparing these definitions with the Euler angles involved in the 321 rotation, we can verify that the names of the Euler angles given for are correct. Specifically, the 3-degree roll corresponds to angle A, the 2-degree pitch corresponds to angle B, and the 1-degree yaw corresponds to the heading angle.(d) To verify that the transformation matrix from wind to body axes is correct, we can multiply the rotation matrix for the 321 rotation by the initial wind axes matrix. If the resulting matrix matches the body axes matrix, then the transformation matrix is correct.